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The electron, proton, and neutron were discovered, respectively, in 1897, 1919, and 1932. The neutron was late to the party, and some physicists felt that it was unnecessary to consider it as fundamental. Maybe it could be explained as simply a proton with an electron trapped inside it. The charges would cancel out, giving the composite particle the correct neutral charge, and the masses at least approximately made sense (a neutron is heavier than a proton).

a. Given that the diameter of a proton is on the order of 10^−15 m, use the Heisenberg uncertainty principle to estimate the trapped electron’s minimum momentum.
b. Find the electron’s minimum kinetic energy.

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Answer:

[tex]5.252\times 10^{-20}\ kgm/s[/tex]

[tex]1.515\times 10^{-9}\ J[/tex]

Explanation:

h = Planck's constant = [tex]6.6\times 10^{-34}\ Js[/tex]

[tex]\Delta P[/tex] = Change in mometum

[tex]\Delta x[/tex] = Change in position = [tex]10^{-15}\ m[/tex]

n = 1

m = Mass of electron = [tex]9.1\times 10^{-31}\ kg[/tex]

From Heisenberg's uncertainty principle we know that

[tex]\Delta x\Delta P=\dfrac{nh}{4\pi}\\\Rightarrow \Delta P=\dfrac{nh}{4\pi\Delta x}\\\Rightarrow \Delta P=\dfrac{1\times 6.6\times 10^{-34}}{4\pi\times 10^{-15}}\\\Rightarrow \Delta P=5.252\times 10^{-20}\ kgm/s[/tex]

The minimum momentum is [tex]5.252\times 10^{-20}\ kgm/s[/tex]

Kinetic energy is given by

[tex]K=\dfrac{\Delta P^2}{2m}\\\Rightarrow K=\dfrac{(5.252\times 10^{-20})^2}{2\times 9.1\times 10^{-31}}\\\Rightarrow K=1.515\times 10^{-9}\ J[/tex]

The kinetic energy is [tex]1.515\times 10^{-9}\ J[/tex]

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